Mathematics in Biological Systems: Understanding Ion Channels

1. Mathematics describes only a tiny part of life, But Mathematics* Creates our Standard of Living *e.g.,Electricity, Computers, Fluid Dynamics, Optics, Structural Mechanics, ….

2. How can we use mathematics to describe biological systems? I believe some biology isPhysics ‘as usual’‘Guess and Check’ But you have to know which biology!

3. + ~30 Å Ion Channelsare theValves of CellsIon Channels are the Main Controllers of Biological Function Ions in Waterare the Selectivity Different Ions carry Different Signals Liquid of Life Na+ Hard Spheres Ca++ Chemical Bonds are lines Surface is Electrical Potential Redis negative (acid) Blueis positive (basic) K+ 3 Å 0.7 nm = Channel Diameter Figure of ompF porin by Raimund Dutzler

4. K+ ~30 Å Ion Channels are Biological Devices Natural nano-valves* for atomic control of biological function Ion channels coordinate contraction of cardiac muscle, allowing the heart to function as a pump Ion channels coordinate contraction in skeletal muscle Ion channels control all electrical activity in cellsIon channels produce signals of the nervous system Ion channels are involved in secretion and absorption in all cells:kidney, intestine, liver, adrenal glands, etc. Ion channels are involved in thousands of diseases and many drugs act on channels Ion channels are proteins whose genes (blueprints) can be manipulated by molecular genetics Ion channels have structures shown by x-ray crystallography in favorable cases *nearly pico-valves: diameter is 400 – 900 picometers

5. Thousands of Molecular Biologists Study Channels every day,One protein molecule at a timeThis number is not an exaggeration.We have sold >10,000 AxoPatch amplifiers Ion Channel Monthly Femto-amps (10-15 A) AxoPatch 200B

6. Channel Structure Does Not Change once the channel is open Amplitude vs. Duration Current vs. time Open Closed Open Amplitude, pA 5 pA 100 ms Open Duration /ms Lowpass Filter = 1 kHz Sample Rate = 20 kHz Typical Raw Single Channel Records Ca2+ Release Channel of Inositol Trisphosphate Receptor: slide and data from Josefina Ramos-Franco. Thanks!

7. Channels are only Holes Why can’t we understand and build them? Where to start? Why not compute all the atoms?

8. Multiscale Issues more later Three Dimensional (106)3 Biological Scales Occur Together so must be Computed Together This may be impossible in simulations Physicists and Engineers rarely try

9. Multiscale Issues It may not be possible to deal accurately with Ratios of Scales of 1011 106 1018 1012 All at Once Physicists and Engineers rarely try! Computational Biology is NOT doing ‘Physics as Usual’

10. Why can’t we understand and build channels? Uncalibrated Simulations will not make devices that actually work Unpopular view because Calibration is Hard Work particularly for Non-Ideal systems with Correlations, Finite Size effects, and Flows

11. Where do we start? Physics ‘As Usual’‘Guess and Check’ Stochastic ‘Derivation’ Later will include biological adaptation of Correlations and Crowded Charge

12. We start with Langevin equations of charged particles Opportunity and Need Simplest stochastic trajectories are Brownian Motion of Charged Particles Einstein, Smoluchowski, and Langevin ignored chargeand therefore do not describe Brownian motion of ions in solutions We useTheory of Stochastic Processesto gofrom Trajectories to Probabilities Once we learn to count Trajectories of Brownian Motion of Charge, we can count trajectories of Molecular Dynamics Schuss, Nadler, Singer, Eisenberg

13. Langevin Equations Positivecation, e.g., p= Na+ Negativeanion, e.g., n= Cl¯ Electric Forcefrom all charges including Permanent charge of Protein, Dielectric Boundary charges,Boundary condition charge Schuss, Nadler, Singer, Eisenberg

14. Electric Force from Poisson Equation Electric Forcefrom all charges including Permanent charge of Protein, Dielectric Boundary charges,Boundary condition charge Excess ‘Chemical’ Force Implicit Solvent‘Primitive’ Model or Primitive Solvent Model Schuss, Nadler, Singer, Eisenberg

15. Nonequilibrium EquilibriumThermodynamics Theory of Stochastic Processes StatisticalMechanics Schuss, Nadler, Singer & Eisenberg Configurations Trajectories Fokker Planck Equation Boltzmann Distribution Finite OPEN System Device Equation Thermodynamics

16. From Trajectories to Probabilities Main Result of Theory of Stochastic Processes Joint probability density of position and velocity satisfies a Fokker Planck equation with Fokker Planck Operator Coordinates are positions and velocities of N particles in 12N dimensional phase space Schuss, Nadler, Singer, Eisenberg

17. Conditional PNP Electric Force depends on Conditional Density of Charge Closures or ApproximationsNeeded Permittivity, Dielectric Coefficient, Charge on Electron ChannelProtein Nernst-Planck gives UNconditional Density of Charge Mass Friction Schuss, Nadler, Singer, Eisenberg

18. Poisson-Nernst-Planck (PNP) Channel Protein Drift-diffusion & Continuity Equation Chemical Potential Chemical Correlations Poisson’s Equation Diffusion Coefficient Number Densities Flux Thermal Energy Thermal Energy Dielectric Coefficient Permittivity Proton charge Valence Valence Proton charge

19. Semiconductor Equations: One Dimensional PNP Poisson’s Equation Drift-diffusion & Continuity Equation Chemical Potential Permanent Charge of Protein Diffusion Coefficient Thermal Energy Cross sectional Area Flux Number Densities Dielectric Coefficient Valence Proton charge valence proton charge

20. Counting at low resolution gives ‘Semiconductor Equations’ Poisson-Nernst-Planck (PNP) Ions are Points in PNP contains only the Correlations of Means Gouy-Chapman, (nonlinear) Poisson-Boltzmann, Debye-Hückel, are siblings with similar resolutionbut at equilibrium, without current or flux of any speciesDevices do not exist at equilibrium

21. How do we check the theory? Compare with Biological Function! Our task is to Discover & Understand, Control & Improve Biological Function Inverse Problem That means Selectivity Ions are not Ideal / Potassium K+ = Na+ Sodium Heinz Engl, Martin Burger

22. + ~30 Å Channels are SelectiveDifferent Ions Carry Different Signals through Different Channels ompF porin Ca++ Na+ K+ 0.7 nm = Channel Diameter 3 Å Diameter mattersDiameter is the Only Difference between K+ and Na+ In ideal solutions K+ = Na+ Flow time scale is 0.1 msec to 1 min Figure of ompF porin by Raimund Dutzler

23. Channels are Selective because Ions are NOT Ideal Potassium K+ = Na+ Sodium / K+ Na+ 3 Å Ideal Electrolytes are Identical if they have the same charge

24. Modelers and Mathematicians, Bioengineers:this is reverse engineering How does theChannel control Selectivity? Inverse ProblemsMany answers are possibleCentral Issue Which answer is right? Core Math Problem has actually been solved using methods for the Inverse Problem of a Blast FurnaceBurger, Eisenberg and Engl (2007) SIAM J Applied Math 67: 960-989

25. How does theChannel control Selectivity? Inverse Problems: many answers possible Central Issue Which answer is right? Key is ALWAYS Large Amount of Data from Many Different Conditions Burger, Eisenberg and Engl (2007) SIAM J Applied Math 67: 960-989

26. Goal: Understand Selectivity well enough toFit Large Amounts of Data from many solutions and concentrationsand to Make a Calcium Channel Atomic Scale Macro Scale

27. Experiments have builtTwo Synthetic Calcium Channels MUTANT ─ Compound Calcium selective Unselective Wild Type As density of permanent charge increases, channel becomes calcium selectiveErev ECa in0.1M1.0 M CaCl2 built by Henk Miedema, Wim Meijberg of BioMade Corp.,Groningen, Netherlands Miedema et al, Biophys J 87: 3137–3147 (2004) Mutants of ompF Porin Designed by Theory Glutathione derivatives Atomic Scale || Macro Scale

28. Comparison with Experiments shows Potassium K+ Sodium Na+ Must include Biological Adaptation! Working Hypothesis Biological Adaptation is Crowded Ions and Side Chains

29. Active Sites of Proteins are Very Charged 7 charges ~ 20M net charge = 1.2×1022 cm-3 liquidWater is 55 Msolid NaCl is 37 M + + + + + - - - - Selectivity Filters and Gates of Ion Channels are Active Sites Physical basis of function OmpF Porin Hard Spheres Na+ Ions are Crowded K+ Ca2+ Na+ Induced Fit of Side Chains K+ 4 Å Figure adapted from Tilman Schirmer

30. Charge Density22 M EC#: Enzyme Commission Number based on chemical reaction catalyzed #AA: Number of residues in the functional pocket MS_A^3: Molecular Surface Area of the Functional Pocket (Units Angstrom^3) CD_MS+: Charge Density (positive) CD_MS-: Charge Density (negative) CD_MSt: Total Charge density Jimenez-Morales, Liang, Eisenberg

31. Example: UDP-N-ACETYLGLUCOSAMINE ENOLPYRUVYL TRANSFERASE (PDB:1UAE) Functional Pocket Molecular Surface Volume: 1462.40 A3 Density Charge: 19.3 Molar (11.3 M+. 8 M-) EC2: TRANSFERASESAverage Charged Density: 19.8 Molar Green: Functional pocket residues Blue: Basic = Positive charged = R+K+H Red: Acid = Negative charged = E + Q Brown URIDINE-DIPHOSPHATE-N- ACETYLGLUCOSAMINE Jimenez-Morales, Liang, Eisenberg

32. Working Hypothesis Biological Adaptation is Crowded Ions and Side Chains Everything interacts

33. Working Hypothesis Interactions in Channels come mostly from Finite Size Effects Chemically Specific Propertiescome from Diameter and Charge learned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

34. Bulk Solutions: Interactions come mostly from Finite Size Effects Chemically Specific Properties of ions (e.g. activity = free energy per mole) are known to come from interactions of their Diameter and Charge and dielectric ‘constant’ of ionic solution Atomic Detail ‘Primitive Implicit Solvent Model’ learned from Doug Henderson, J.-P. Hansen, Stuart Rice, among others…Thanks!

35. Ions in Water are the Liquid of Life They are not ideal solutions Everything Interacts with Everything For Modelers and Mathematicians Tremendous Opportunity for Applied MathematicsChun Liu’s Energetic Variational Principle EnVarA

36. Variational Principles Deal with Interactions Consistently and Automatically Chun Liu, with Yunkyong Hyon, and Bob Eisenberg EnVarA New Component(or Scale)implies New Field Equations (Euler Lagrange) by Algebra Alone No new Assumptions

37. Energetic Variational ApproachEnVarAChun Liu, Rolf Ryham, Yunkyong Hyon, and Bob Eisenberg Mathematicians and Modelers: two different ‘partial’ variations written in one framework, using a ‘pullback’ of the action integral CompositeVariational Principle Action Integral, after pullback Rayleigh Dissipation Function Euler Lagrange Equations Field Theory of Ionic Solutions that allows boundary conditions and flow and deals with Interactions of Components self-consistently

38. Variational Analysis of Ionic Solution EnVarA Generalization of Chemical Free Energy Dielectric Coefficient from Poisson Eq. Number Densities Lagrange Multiplier Eisenberg, Hyon, and Liu

39. EnVarA Dissipation Principle for Ions Hard Sphere Terms Number Density Thermal Energy time Permanent Charge of protein valence proton charge cinumber density; thermal energy; Di diffusion coefficient; n negative; p positive; zivalence Eisenberg, Hyon, and Liu

40. Field Equations with Lennard Jones Spheres Non-equilibriium variational field theory EnVarA Nernst Planck Diffusion Equation for number density cnof negative n ions; positive ions are analogous Diffusion Coefficient Coupling Parameters Thermal Energy Permanent Charge of Protein Ion Radii Number Densities Poisson Equation Dielectric Coefficient valence proton charge Eisenberg, Hyon, and Liu

41. Energetic Variational ApproachEnVarA across biological scales: molecules, cells, tissuesdeveloped by Chun Liuwith (1) Hyon, Eisenberg Ions in Channels (2) Bezanilla, Hyon, Eisenberg Conformation Change of Voltage Sensor (3) Ryham, Eisenberg, Cohen Virus fusion to Cells (4) Mori, Eisenberg Water flow in Tissues Multiple Scales creates a newMultiscale Field Theory of Interacting Components that allows boundary conditions and flow and deals with Ions in solutions self-consistently

42. Energetic Variational Approach developed by Chun Liu Preliminary Results and Provocations

43. Eisenberg, Hyon, and Liu

44. Nonequilibrium Computations with Variational Field Theory EnVarA Current Voltage Time Curves Binding Curves

45. Energetic VariationalApproachEnVarA New mechanisms* (e.g., active transport) can be added *if they define an energy and its variation Energy defined by simulations or theories or experiments is OK Full micro/macro treatment is needed for an Atomic Model, with closure, as in liquid crystals Eisenberg, Hyon, and Liu

46. Back to the Calcium Channel Then, the Sodium Channel

47. Selectivity FilterCrowded with Charge Selectivity Filter O½ Wolfgang Nonner L type Ca Channel + ++ “Side Chains”

48. large mechanical forces

49. Multiscale Analysis at Equilibrium Solved with Metropolis Monte Carlo MMC Simulates Location of Ionsboth the mean and the variance ProducesEquilibrium Distribution of location of Ions and ‘Side Chains’ MMC yields Boltzmann Distribution with correct Energy, Entropy and Free Energy Other methodsgive nearly identical results: Equilibrium MultiscaleMSA (mean spherical approximation SPM (primitive solvent model) DFT (density functional theory of fluids), Non-equilibrium Multiscale DFT-PNP (Poisson Nernst Planck) EnVarA…. (Energy Variational Approach) etc

50. Metropolis Monte Carlo Simulates Location of Ionsboth the mean and the variance Details: • Start with Configuration A, with computed energy EA • Move an ion to location B, with computed energy EB • If spheres overlap, EB → ∞ and configuration is rejected • If spheres do not overlap, EB→ 0 and configuration is accepted • If EB < EA: accept new configuration. • If EB > EA : accept new configuration with probability Key idea MMC chooses configurations with a Boltzmann probability and weights them evenly instead of choosing them from uniform distribution and then weighting them with exp(−E/k BT)